<?php
/**
 * <https://y.st./>
 * Copyright © 2018 Alex Yst <mailto:copyright@y.st>
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**/

$xhtml = array(
	'<{title}>' => 'Inverses and composites',
	'<{subtitle}>' => 'Written in <span title="College Algebra">MATH 1201</span> by <a href="https://y.st./">Alex Yst</a>, finalised on 2018-02-28',
	'<{copyright year}>' => '2018',
	'takedown' => '2017-11-01',
	'<{body}>' => <<<END
<section id="problem0">
	<h2>Problem 0</h2>
	<p>
		f ○ g can be rewritten as f(g(x)).
		That means that <strong>f ○ g is equal to 2x<sup>2</sup> + 1</strong>.
		Thankfully, this is already in its reduced form as soon as we plug it in.
	</p>
	<p>
		g ○ f can be rewritten as g(f(x)).
		That makes it equal to (2x + 1)<sup>2</sup>.
		Unlike the previous arrangement, this one will need to be $a[FOIL]ed.
		Doing that, we see <strong>g ○ f is equal to 4x<sup>2</sup> + 4x + 1</strong>.
	</p>
</section>
<section id="problem1">
	<h2>Problem 1</h2>
	<p>
		To undo a function, we must apply the reverse operations of that function, in reverse order.
		This means we actually need to apply the order of operations in reverse, which can be done using parentheses.
		The inverse of f(x) = 2x + 3 is therefore <strong>f(x) = (x - 3) ÷ 2</strong>.
	</p>
</section>
<section id="problem2">
	<h2>Problem 2</h2>
	<p>
		This function is a little more complex, due to its artificially-restricted domain.
		To begin, we can undo the operations of the function in reverse order as before: f(x) = ±√(x - 3).
		Then, we need to look at the domain restriction.
		X has to be less than zero.
		For starters, that means we can convert the plus-or-minus to simply a negative.
		Second, we need to look at what that limits the original equation&apos;s output to and in which direction the output goes as we travel away from the limit.
		Plugging in zero to the initial function, we see the limit for the inverse needs to be either x &lt; 3 or x &gt; 3.
		If we plug in negative one, we get an output of four, so as the input decreases, the output increases.
		That means the input to the inverse needs to be able to rise, but not fall to or below three.
		The inverse of f(x) = x<sup>2</sup> + 3, x &lt; 0 is therefore <strong>f(x) = -√(x - 3), x &gt; 3</strong>.
	</p>
</section>
END
);

